THE PBDW method import sys !{sys.executable} -m pip install numpy !{sys.executable} -m pip install matplotlib !{sys.executable} -m pip install scikit-fem Let us present one method that combines model order reduction and a data assimilation problem: the Parametrized-Background Data-Weak method (PBDW). $$ G^{bk,\mu}(u^{bk}(\mu)) = 0.$$$$ \mathcal{M}^{bk} = \{u^{bk}(\mu) : \mu \in \mathcal{P}^{bk}\}.$$The PBDW formulation integrates the parameterized mathematical model $G^{pb}$ and $M$ experimental observations associated with a parameter configuration $\mu^{true}$ to estimate the true field $u^{true}(\mu^{true})$ as well as any desired output $l_{out}(u^{true}(\mu^{true}\ )) \in \mathbb{R}$ for given output functional $l_{out}$. We intend that $\lVert u^{true}(\mu^{true})-u^{bk}(\mu^{true}) \rVert$ is small (i.e. that our model represents our data observations well).
Dec 13, 2024
THE POD-I METHOD Script by Elise Grosjean (ENSTA-Paris, elise.grosjean@ensta-paris.fr) import sys !{sys.executable} -m pip install numpy !{sys.executable} -m pip install matplotlib !{sys.executable} -m pip install scikit-fem !{sys.executable} -m pip install scikit-learn Let us consider a parameterized problem. Having (previously computed) solutions for different parameter values, we aim at approaching much faster the solution associated to a new parameter value. The Galerkin-Proper Orthogonal Decomposition (POD) algorithm is decomposed in two parts:
Dec 13, 2024
THE POD-GALERKIN METHOD Script by Elise Grosjean (ENSTA-Paris, elise.grosjean@ensta-paris.fr) import sys !{sys.executable} -m pip install numpy !{sys.executable} -m pip install matplotlib !{sys.executable} -m pip install scikit-fem Let us consider a parameterized problem. Having (previously computed) solutions for different parameter values, we aim at approaching much faster the solution associated to a new parameter value. The Galerkin-Proper Orthogonal Decomposition (POD) algorithm is decomposed in two parts:
Dec 13, 2024
THE PBDW method import sys !{sys.executable} -m pip install numpy !{sys.executable} -m pip install matplotlib !{sys.executable} -m pip install scikit-fem Let us present one method that combines model order reduction and a data assimilation problem: the Parametrized-Background Data-Weak method (PBDW). $$ G^{bk,\mu}(u^{bk}(\mu)) = 0.$$$$ \mathcal{M}^{bk} = \{u^{bk}(\mu) : \mu \in \mathcal{P}^{bk}\}.$$The PBDW formulation integrates the parameterized mathematical model $G^{pb}$ and $M$ experimental observations associated with a parameter configuration $\mu^{true}$ to estimate the true field $u^{true}(\mu^{true})$ as well as any desired output $l_{out}(u^{true}(\mu^{true}\ )) \in \mathbb{R}$ for given output functional $l_{out}$. We intend that $\lVert u^{true}(\mu^{true})-u^{bk}(\mu^{true}) \rVert$ is small (i.e. that our model represents our data observations well).
Dec 13, 2024
THE PBDW method import sys !{sys.executable} -m pip install numpy !{sys.executable} -m pip install matplotlib !{sys.executable} -m pip install scikit-fem Let us present one method that combines model order reduction and a data assimilation problem: the Parametrized-Background Data-Weak method (PBDW). $$ G^{bk,\mu}(u^{bk}(\mu)) = 0.$$$$ \mathcal{M}^{bk} = \{u^{bk}(\mu) : \mu \in \mathcal{P}^{bk}\}.$$The PBDW formulation integrates the parameterized mathematical model $G^{pb}$ and $M$ experimental observations associated with a parameter configuration $\mu^{true}$ to estimate the true field $u^{true}(\mu^{true})$ as well as any desired output $l_{out}(u^{true}(\mu^{true}\ )) \in \mathbb{R}$ for given output functional $l_{out}$. We intend that $\lVert u^{true}(\mu^{true})-u^{bk}(\mu^{true}) \rVert$ is small (i.e. that our model represents our data observations well).
Dec 13, 2024
THE NIRB TWO-GRID METHOD Script by Elise Grosjean (elise.grosjean@ensta-paris.fr) import sys !{sys.executable} -m pip install numpy !{sys.executable} -m pip install matplotlib !{sys.executable} -m pip install scikit-fem Let us consider a parameterized problem. The NIRB two-grid method follows the same ideas as in the Snapshot Proper Orthogonal Decomposition (POD) algorithm and is decomposed in two parts:
Dec 13, 2024
The Empiral Interpolation Method (EIM) Script by Elise Grosjean (ENSTA-Paris, elise.grosjean@ensta-paris.fr) import sys !{sys.executable} -m pip install numpy !{sys.executable} -m pip install matplotlib !{sys.executable} -m pip install scikit-fem The EIM build a linear combination of fully determined solutions from basis functions $(q_i)_{i=1,...,M}$ depending on some interpolating points (called “magic points”) and some values $(\alpha_i)_{i=1,...,M}$ relying on certain instances of the parameter $\nu$, selected within the algorithm.
Dec 13, 2024